$\mathrm{GL}(2)$-structures in dimension four, $H$-flatness and integrability

TL;DR

This paper demonstrates that torsion-free four-dimensional $ ext{GL}(2)$-structures are flat modulo a specific subgroup transformation and that the associated PDE system is integrable via a dispersionless Lax pair.

Contribution

It establishes the flatness of torsion-free 4D $ ext{GL}(2)$-structures up to a subgroup and proves the integrability of the related PDE system.

Findings

Torsion-free 4D $ ext{GL}(2)$-structures are flat up to an $H$-transformation.

The PDE system associated with these structures admits a dispersionless Lax pair.

The subgroup $H$ is isomorphic to a semidirect product of $H_3( ext{R})$ and $ ext{R}$.

Abstract

We show that torsion-free four-dimensional $\mathrm{GL}(2)$-structures are flat up to a coframe transformation with a mapping taking values in a certain subgroup $H\subset\mathrm{SL}(4,\mathbb{R})$ which is isomorphic to a semidirect product of the three-dimensional continuous Heisenberg group $H_3(\mathbb{R})$ and the Abelian group $\mathbb{R}$. In addition, we show that the relevant PDE system is integrable in the sense that it admits a dispersionless Lax-pair.